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Wagner model

Wagner model is a rheological model developed for the prediction of the viscoelastic properties of polymers. It might be considered as a simplified practical form of the Bernstein-Kearsley-Zapas model. The model was developed by German rheologist Manfred Wagner.

For the isothermal conditions the model can be written as:

\mathbf{\sigma}(t) = -p \mathbf{I} + \int_{-\infty}^{t} M(t-t')h(I_1,I_2)\mathbf{B}(t')\, dt'


  • \mathbf{\sigma}(t) is the stress tensor as function of time t,
  • p is the pressure
  • \mathbf{I} is the unity tensor
  • M is the memory function showing, usually expressed as a sum of exponential terms for each mode of relaxation:
M(x)=\sum_{k=1}^m \frac{g_i}{\theta_i}\exp(\frac{-x}{\theta_i}), where for each mode of the relaxation, gi is the relaxation modulus and θi is the relaxation time;
  • h(I1,I2) is the strain damping function that depends upon the first and second invariants of Finger tensor \mathbf{B}.

The strain damping function is usually written as:

h(I_1,I_2)=m^*exp(-n_1 \sqrt{I_1-3})+(1-m^*)exp(-n_2 \sqrt{I_2-3}),

The strain hardening function equal to one, then the deformation is small and approaching zero, then the deformations are large.

The Wagner equation can be used in the non-isothermal cases by applying time-temperature shift factor.


  • M.H. Wagner Rheologica Acta, v.15, 136 (1976)
  • M.H. Wagner Rheologica Acta, v.16, 43, (1977)
  • B. Fan, D. Kazmer, W. Bushko, Polymer Engineering and Science, v44, N4 (2004)
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Wagner_model". A list of authors is available in Wikipedia.
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